Floor Vibration — AISC Design Guide 11 Walking Excitation Check

Floor vibration is a serviceability problem, not a strength problem. A floor can pass every strength and deflection check but still be unacceptable to occupants if it bounces perceptibly when someone walks across it. Steel-framed composite floors are particularly susceptible because they combine light weight with long spans, producing low natural frequencies that coincide with the frequency range of human walking (1.6–2.2 Hz for normal pace). AISC Design Guide 11, 2nd Edition provides the standard assessment method for walking-induced vibration.

The DG11 criterion

The fundamental check compares the peak acceleration from walking excitation to a tolerance limit:

ap/g = P0 × exp(-0.35 × fn) / (beta × W) ≤ ao/g

Where:

• ap/g = peak acceleration as a fraction of gravity
• P0 = constant force representing a walking person = 65 lb (0.29 kN)
• fn = fundamental natural frequency of the floor bay (Hz)
• beta = damping ratio (fraction of critical damping)
• W = effective weight of the floor panel participating in vibration (lb)
• ao/g = acceleration tolerance limit

Acceleration tolerance limits

For offices, the floor passes if ap/g ≤ 0.005 (0.5% of gravity). This is extremely stringent — the human body is remarkably sensitive to vertical acceleration at frequencies between 4 and 8 Hz.

Damping ratio values

Most office designs use beta = 0.025 (ceiling + ductwork, but open-plan with few partitions).

Calculating natural frequency (fn)

The natural frequency depends on the combined stiffness and mass of the beam-girder system. For a simply supported composite beam:

fn = 0.18 × sqrt(g / delta_j)    where delta_j is the midspan deflection under sustained load (inches)

The sustained load includes the slab self-weight, superimposed dead load, and 10–25% of the design live load (representing the "normally present" furniture and occupants — not the full code live load).

For a two-way system (beams supported on girders), the combined frequency is:

fn = 0.18 × sqrt(g / (delta_j + delta_g))

Where delta_j = beam midspan deflection and delta_g = girder midspan deflection (both under sustained loads). The girder deflection should be calculated at the beam connection point.

Worked example — office floor vibration check

Given: W16x26 composite beams at 10 ft o.c., span 35 ft. W24x55 composite girder, span 30 ft. 3" deck + 3.25" LW concrete (unit weight = 110 pcf). SDL = 15 psf, partition load = 10 psf. Office occupancy.

Step 1 — Beam frequency: Sustained load on beam: slab = 42 psf × 10 ft = 420 plf. SDL = 15 × 10 = 150 plf. Partitions = 10 × 10 = 100 plf. Beam = 26 plf. Total w = 696 plf = 0.058 kli. I_composite = 935 in4 (from composite section properties with full Itr). delta_j = 5 × w × L4 / (384 × E × I) = 5 × 0.058 × (420)4 / (384 × 29000 × 935) = 0.87 in. fn,beam = 0.18 × sqrt(386/0.87) = 3.79 Hz.

Step 2 — Girder frequency: Girder carries beam reactions at third points. Equivalent uniform load for two-point loading pattern. I_girder,composite = 3200 in4. Girder deflection delta_g = 0.35 in (calculated from beam reactions). fn,girder = 0.18 × sqrt(386/0.35) = 5.98 Hz.

Step 3 — Combined frequency: fn = 0.18 × sqrt(386 / (0.87 + 0.35)) = 0.18 × sqrt(316) = 3.20 Hz.

Step 4 — Effective weight: W_beam = wj × Bj × Lj, where Bj = beam panel width (affected by beam spacing and span). Per DG11 Section 4.1, Bj = Cj × (Ds/Dj)^0.25 × Lj, where Ds = slab stiffness, Dj = beam transformed stiffness per unit width. Typical W ≈ 90,000 lb for this bay.

Step 5 — Peak acceleration: beta = 0.025 (office, ceiling, no full-height partitions). ap/g = 65 × exp(-0.35 × 3.20) / (0.025 × 90000) = 65 × 0.326 / 2250 = 0.0094 = 0.94% g > 0.5% g limit — FAILS.

Resolution: Increase beam size to W18x35 (higher stiffness → higher fn → lower ap/g) or add girder stiffness. This is a classic case where vibration, not strength or deflection, controls the beam size.

Code comparison

AISC Design Guide 11 (USA): The primary reference for US practice. Uses the resonance acceleration method with P0 = 65 lb. Covers walking, rhythmic activity, and sensitive equipment. Does not have the force of a code requirement but is universally adopted by specification.

SCI P354 (UK/Europe): Uses a response factor approach based on BS 6472 and EN 1990 Annex A1.4.4. The walking force model is more detailed (Fourier series with harmonic components). The acceptance criterion is expressed as a response factor R (multiplier of the base perception threshold), typically R ≤ 8 for offices. SCI P354 generally produces more conservative results than DG11 for long-span floors.

AS 3600-2018 / CCAA T53 (Australia): No standalone vibration design guide equivalent to DG11. Australian practice typically references DG11 directly or uses the Concrete Centre guide (CCIP-016) for composite floors. AS 3600 Section 2.3.4 requires consideration of vibration for prestressed floors but provides no specific method for steel composite floors.

Common mistakes engineers make

1. Using full design live load to calculate fn. The natural frequency should be calculated using the mass actually present on the floor during normal use — typically 10–25% of the code live load (11 psf for a 50 psf office). Using the full 50 psf live load artificially increases the mass, lowers fn, and produces an unconservative result (lower fn means higher ap/g from the exponential).

2. Checking only the beam and ignoring the girder flexibility. The combined system frequency is always lower than the individual beam or girder frequency. A beam with fn = 7 Hz on a flexible girder with fn = 5 Hz gives a combined fn of approximately 4.1 Hz — potentially below the acceptable range.

3. Assuming that a stiffer floor is always better. Increasing stiffness raises fn, which reduces the walking excitation (due to the exponential decay). However, increasing stiffness also reduces deflection delta_j, which reduces the effective weight W. If W drops proportionally, the net effect on ap/g can be neutral. Both frequency and mass must be evaluated together.

4. Ignoring adjacent bays. DG11 calculates an effective panel width that can extend beyond the bay directly being loaded. If adjacent bays have different stiffness or mass (e.g., a stairwell opening, a setback, or a change in beam size), the effective weight calculation must account for the actual boundary conditions.

AISC Design Guide 11 — detailed walking vibration criteria

AISC Design Guide 11 (DG11), 2nd Edition, establishes two parallel criteria for evaluating floor vibration due to walking excitation: (1) a frequency criterion and (2) an acceleration criterion. Both must be satisfied for the floor to be acceptable for its intended occupancy.

Natural frequency limits

DG11 recommends minimum fundamental frequencies based on occupancy to avoid resonance with walking excitation harmonics. Walking produces a periodic force with fundamental frequency f_step between 1.6 and 2.2 Hz. The first four harmonics occur at f_step, 2 x f_step, 3 x f_step, and 4 x f_step. For a 2.0 Hz walking pace, the fourth harmonic is at 8.0 Hz. A floor with a natural frequency below 8 Hz can be excited by one of the first four walking harmonics.

The 5 Hz recommended minimum for offices is not a hard pass/fail threshold — it is a benchmark. Floors with fn between 3 and 5 Hz may still be acceptable if the acceleration check passes, but the floor is more sensitive to excitation and requires careful evaluation.

Acceleration limits (ao/g)

The acceleration limit is the controlling criterion for most designs. DG11 specifies the following peak acceleration tolerances as a fraction of gravitational acceleration (g = 386 in/s^2):

The residential limit of 0.2% g is notably more stringent than the office limit. This is because residential occupants are typically alone, stationary, and in a quiet environment where even minimal floor motion is perceptible and annoying. The difference between 0.2% g and 0.5% g often dictates whether a floor system is acceptable for residential conversion of commercial buildings.

Effective weight method (W_eff)

The effective weight W (also written W_eff) represents the total mass of floor area that participates in the vibration mode. It is NOT simply the dead load of one bay — it depends on the beam spacing, slab stiffness, and the mode shape of the vibrating panel.

Fundamental formula

For a single beam acting alone (one-way system):

W_eff = w x L x B

Where:

• w = sustained load per unit length (lb/ft), including slab self-weight, SDL, and 10-25% of live load
• L = beam span (ft)
• B = effective floor panel width (ft), which accounts for the tributary area that vibrates with the beam

Effective panel width (B)

The effective panel width B is calculated per DG11 Section 4.1:

B = C x (Ds / Dj)^0.25 x L

Where:

• C = mode shape coefficient (1.0 for simple span, 0.7 for continuous beam)
• Ds = slab flexural stiffness per unit width (k-in)
• Dj = beam flexural stiffness per unit width (k-in/ft)
• L = beam span (ft)

For typical composite floors with 3-inch deck and lightweight concrete:

• B ranges from 0.4 x L to 0.8 x L depending on beam spacing and slab thickness
• Closer beam spacing produces a larger effective width (more of the slab moves with the beam)
• Wider beam spacing produces a smaller effective width

Natural frequency from deflection

The fundamental frequency of a simply supported beam can be estimated directly from its midspan deflection under sustained load:

f_n = 0.18 x sqrt(g / delta)

Where:

• g = 386 in/s^2 (gravitational acceleration)
• delta = midspan deflection under sustained load (inches)
• f_n = fundamental frequency (Hz)

This formula is derived from the beam vibration equation f_n = (pi / 2) x sqrt(EI / (wL^4)) / (2pi), rearranged using delta = 5wL^4 / (384EI). The 0.18 coefficient combines all constants.

Important: The deflection delta must be computed using the transformed composite section (fully composite, with the full modular ratio n = E_steel / E_concrete). Do NOT use the cracked or non-composite section properties.

Common problem spans and remedial fixes

Span ranges where vibration problems are common

Remedial strategies

1. Increase beam depth or weight. Going from a W16x31 to a W18x35 increases the composite moment of inertia by approximately 30-40%, which raises fn and reduces ap/g. This is the simplest and most common fix. The cost premium is typically 10-15% of the steel package for the affected beams.

2. Reduce beam spacing. Changing from 10 ft on center to 7.5 ft on center increases the effective weight (more slab mass moves with each beam) and raises the system frequency. This adds beams but may allow lighter individual beam sizes.

3. Add damping. Damping is the most cost-effective parameter to improve. Adding full-height partitions increases beta from 0.025 to 0.04-0.05, reducing ap/g proportionally. Tuned mass dampers (TMDs) can be installed on problematic floors: a TMD adds a counter-oscillating mass that absorbs vibration energy. TMDs cost $5,000-$15,000 per bay but avoid structural modifications.

4. Increase girder stiffness. If the girder frequency fn,girder is the bottleneck (because it is lower than fn,beam), upsizing the girder can be more effective than upsizing every beam. A single girder change fixes all bays that frame into it.

5. Use cambered beams with lighter composite action. Paradoxically, partial composite action can sometimes help by making the beam more flexible (lower fn), which shifts the natural frequency away from a walking harmonic. However, this approach requires careful analysis because lower fn also increases the exponential term in the ap/g equation.

Worked example — W16x31 at 30 ft span for office walking vibration

Given: W16x31 composite beam, span L = 30 ft, spacing s = 8 ft on center. 2-inch deck with 3.5-inch lightweight concrete topping (110 pcf, total slab thickness = 5.5 in). Superimposed dead load SDL = 15 psf. Partition load = 10 psf. Office occupancy (ao/g = 0.5%). Ceiling and ductwork below, open-plan (beta = 0.025).

Step 1 — Sustained load:

Slab weight = 5.5 in x 110 pcf / 12 x (1 - deck rib voids, approx. 0.70 factor) = 5.5 x 110 / 12 x 0.70 = 35.5 psf. (Alternatively, from deck manufacturer tables: 42 psf for this configuration.)

w_slab  = 42 psf x 8 ft = 336 plf
w_SDL   = 15 psf x 8 ft = 120 plf
w_part  = 10 psf x 8 ft =  80 plf
w_beam  = 31 plf
w_total = 567 plf = 0.0473 klf

Step 2 — Composite section properties:

For W16x31 with 5.5 in LW concrete slab (f'c = 4 ksi, n = 22): From transformed section analysis: I_composite = 780 in^4 (fully composite, transformed).

Step 3 — Beam midspan deflection under sustained load:

delta_j = 5 x w x L^4 / (384 x E x I)
delta_j = 5 x 0.0473 x (360)^4 / (384 x 29000 x 780)
delta_j = 5 x 0.0473 x 1.680e10 / (8.671e9)
delta_j = 3.970e9 / 8.671e9
delta_j = 0.458 in

Step 4 — Natural frequency of beam:

fn,beam = 0.18 x sqrt(386 / 0.458)
fn,beam = 0.18 x sqrt(842.8)
fn,beam = 0.18 x 29.03
fn,beam = 5.23 Hz

This is above the 3.0 Hz minimum but below the 5.0 Hz recommended threshold for offices. Proceed to acceleration check.

Step 5 — Effective weight:

Using DG11 simplified approach for a single beam: Effective panel width B = 0.6 x L = 0.6 x 30 = 18 ft (typical for 8 ft spacing with LW concrete slab).

W = w x B = 567 plf x 18 ft = 10,206 lb

Note: For a two-way system with girders, the combined effective weight accounts for both beam and girder panels. Here we assume the girder is stiff (fn,girder >> fn,beam), so the beam frequency dominates.

Step 6 — Peak acceleration check:

ap/g = P0 x exp(-0.35 x fn) / (beta x W)
ap/g = 65 x exp(-0.35 x 5.23) / (0.025 x 10,206)
ap/g = 65 x exp(-1.831) / 255.2
ap/g = 65 x 0.1603 / 255.2
ap/g = 10.42 / 255.2
ap/g = 0.0409 = 4.09% g

Result: 4.09% g >> 0.5% g limit. FAILS by a wide margin.

Step 7 — Remedy — upsize to W21x44:

Repeating with W21x44: I_composite = 1,620 in^4.

w_total = 567 - 31 + 44 = 580 plf = 0.0483 klf

delta_j = 5 x 0.0483 x (360)^4 / (384 x 29000 x 1620)
delta_j = 5 x 0.0483 x 1.680e10 / (1.806e10)
delta_j = 0.224 in

fn = 0.18 x sqrt(386 / 0.224) = 0.18 x sqrt(1723) = 0.18 x 41.51 = 7.47 Hz

W = 580 x 18 = 10,440 lb

ap/g = 65 x exp(-0.35 x 7.47) / (0.025 x 10,440)
ap/g = 65 x exp(-2.615) / 261.0
ap/g = 65 x 0.0731 / 261.0
ap/g = 4.75 / 261.0
ap/g = 0.0182 = 1.82% g

Still fails. This illustrates the difficulty of solving floor vibration by simply upsizing beams — a common frustration in practice.

Step 8 — Remedy — reduce spacing to 6 ft on center with W16x36:

At 6 ft spacing, the effective panel width increases: B = 0.7 x 30 = 21 ft (closer spacing = wider effective panel).

w_total = 42x6 + 15x6 + 10x6 + 36 = 252 + 90 + 60 + 36 = 438 plf = 0.0365 klf

I_composite (W16x36) = 870 in^4

delta_j = 5 x 0.0365 x (360)^4 / (384 x 29000 x 870) = 0.276 in

fn = 0.18 x sqrt(386 / 0.276) = 0.18 x sqrt(1399) = 0.18 x 37.4 = 6.73 Hz

W = 438 x 21 = 9,198 lb

ap/g = 65 x exp(-0.35 x 6.73) / (0.025 x 9198)
ap/g = 65 x 0.0953 / 230.0
ap/g = 6.19 / 230.0
ap/g = 0.0269 = 2.69% g

Still above 0.5%. The most effective remedy for a 30 ft span with office occupancy is typically a combination of: deeper beams (W21 or W24), tighter spacing, and increased damping from full-height partitions (beta = 0.05).

Step 9 — Combined remedy — W21x50 at 7 ft o.c. with partitions (beta = 0.05):

w_total = 42x7 + 15x7 + 10x7 + 50 = 294 + 105 + 70 + 50 = 519 plf = 0.0433 klf

I_composite (W21x50) = 1,860 in^4

delta_j = 5 x 0.0433 x (360)^4 / (384 x 29000 x 1860) = 0.184 in

fn = 0.18 x sqrt(386 / 0.184) = 0.18 x sqrt(2098) = 0.18 x 45.8 = 8.24 Hz

B = 0.65 x 30 = 19.5 ft
W = 519 x 19.5 = 10,121 lb

ap/g = 65 x exp(-0.35 x 8.24) / (0.05 x 10121)
ap/g = 65 x 0.0563 / 506.1
ap/g = 3.66 / 506.1
ap/g = 0.0072 = 0.72% g

With beta = 0.05 (full-height partitions), the result drops to 0.72% g — still marginally above the 0.5% limit. Adding the girder flexibility would reduce fn further. This demonstrates that 30 ft spans in office construction with lightweight concrete are at the practical limit for conventional composite beam construction and may require special measures (heavier beams, tuned mass dampers, or normal-weight concrete).

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• How to Verify Calculations
• composite beam design
• beam deflection limits reference
• structural beam formulas reference
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• Castellated Beam
• Floor Systems

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